Shear-driven parametric instability in a precessing sphere

The present numerical study aims at shedding light on the mechanism underlying the precessional instability in a sphere. Precessional instabilities in the form of parametric resonance due to topographic coupling have been reported in a spheroidal geometry both analytically and numerically. We show that such parametric resonances can also develop in spherical geometry due to the conical shear layers driven by the Ekman pumping singularities at the critical latitudes. Scaling considerations lead to a stability criterion of the form, $|P_o|>O(E^{4/5})$, where $P_o$ represents the Poincar\'e number and $E$ the Ekman number. The predicted threshold is consistent with our numerical simulations as well as previous experimental results. When the precessional forcing is supercriticial, our simulations show evidence of an inverse cascade, i.e. small scale flows merging into large scale cyclones with a retrograde drift. Finally, it is shown that this instability mechanism may be relevant to precessing celestial bodies such as the Earth and Earth's moon.